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Central limit theorems and bootstrap in high dimensions

Author

Listed:
  • Victor Chernozhukov

    () (Institute for Fiscal Studies and MIT)

  • Denis Chetverikov

    () (Institute for Fiscal Studies and UCLA)

  • Kengo Kato

    (Institute for Fiscal Studies)

Abstract

In this paper, we derive central limit and bootstrap theorems for probabilities that centered high-dimensional vector sums hit rectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for the probabilities that a root-n rescaled sample average of Xi is in A, where X1,..., Xnare independent random vectors in Rp and A is a rectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if p=pn-> infinity and p>>n; in particular, p can be as large as O(e^(Cn^c)) for some constants c,C>0. The result holds uniformly over all rectangles, or more generally, sparsely convex sets, and does not require any restrictions on the correlation among components of Xi. Sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend nontrivially only on a small subset of their arguments, with rectangles being a special case.

Suggested Citation

  • Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2016. "Central limit theorems and bootstrap in high dimensions," CeMMAP working papers CWP39/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  • Handle: RePEc:ifs:cemmap:39/16
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    File URL: https://www.ifs.org.uk/uploads/cemmap/wps/cwp391616.pdf
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    References listed on IDEAS

    as
    1. Yosef Rinott & Vladimir Rotar, 2000. "Normal approximations by Stein's method," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(1), pages 15-29.
    2. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2012. "Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors," Papers 1212.6906, arXiv.org, revised Jan 2018.
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    Cited by:

    1. Victor Chernozhukov & Denis Chetverikov & Kengo Kato, 2013. "Testing Many Moment Inequalities," CeMMAP working papers CWP65/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Fabian Dunker & Konstantin Eckle & Katharina Proksch & Johannes Schmidt-Hieber, 2017. "Tests for qualitative features in the random coefficients model," Papers 1704.01066, arXiv.org, revised Mar 2018.
    3. Denis Chetverikov & . ., 2016. "On cross-validated Lasso," CeMMAP working papers CWP47/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Victor Chernozhukov & Denis Chetverikov & Mert Demirer & Esther Duflo & Christian Hansen & Whitney K. Newey, 2016. "Double machine learning for treatment and causal parameters," CeMMAP working papers CWP49/16, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Chernozhukov, Victor & Chetverikov, Denis & Kato, Kengo, 2016. "Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3632-3651.

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